An analysis on neural dynamics with saturated sigmoidal functions

We propose a unified approach to study the relation between the set of saturated attractors and the set of system parameters of the Hopfield model, Linsker's model, and the dynamic link network (DLN), which use saturated sigmoidal functions in its dynamics of the state or weight. The key point for this approach is to rigorously derive a necessary and sufficient condition to test whether a given saturated state (in the Hopfield model) or weight vector (in Linsker's model and the DLN) is stable or not for any given set of system parameters, and used this to determine the complete regime in the parameter space over which the given state or weight is stable. Our approach allows us to give an exact characterization between the parameters and the capacity in the Hopfield model; to generalize our previous results on Linsker's network and the DLN; to have a better understanding of the underlying mechanism among these models. The method reported here could be adopted to analyze a variety of models in the field of the neural networks.